Question

If the dimensions of length are expressed as \( G^x C^y h^z \), where \( G \), \( C \), and \( h \) are the universal gravitational constant, speed of light, and Planck's constant respectively, then:

• A. \( x = \frac{1}{2}, y = \frac{1}{2} \)
• B. \( x = \frac{1}{2}, z = \frac{1}{2} \)
• C. \( y = \frac{1}{2}, z = \frac{3}{2} \)

Hint:

In problems involving dimensional analysis, express each physical constant in terms of its fundamental dimensions (mass, length, time) and solve for the exponents that satisfy the dimensional consistency of the equation.

Answer 1

The Correct Option is C

We are given that the dimensions of length are expressed as \( G^x C^y h^z \), where \( G \), \( C \), and \( h \) are the universal gravitational constant, speed of light, and Planck's constant, respectively. We need to find the values of \( x \), \( y \), and \( z \) such that this expression represents length. Step 1: Analyzing the Dimensions of \( G \), \( C \), and \( h \)
We first write the dimensions of each of the physical constants \( G \), \( C \), and \( h \). - The universal gravitational constant \( G \) has the dimensions of: \[ [G] = \frac{M^{-1} L^3 T^{-2}}{\text{dimension of mass (M)}} = M^{-1} L^3 T^{-2} \] - The speed of light \( C \) has the dimensions of: \[ [C] = \frac{L}{T} \] - Planck's constant \( h \) has the dimensions of: \[ [h] = M L^2 T^{-1} \] Step 2: Dimensions of \( G^x C^y h^z \)
Now, we need to find the dimensions of \( G^x C^y h^z \). Using the dimensions of each constant, we have: \[ [G^x C^y h^z] = (M^{-1} L^3 T^{-2})^x \times \left(\frac{L}{T}\right)^y \times (M L^2 T^{-1})^z \] This simplifies to: \[ = M^{-x} L^{3x} T^{-2x} \times L^y T^{-y} \times M^z L^{2z} T^{-z} \] Now, group the terms involving \( M \), \( L \), and \( T \): \[ = M^{-x + z} L^{3x + y + 2z} T^{-2x - y - z} \] For this expression to have the dimensions of length, the powers of \( M \), \( L \), and \( T \) must match the dimensions of length, which are \( [L] = L^1 M^0 T^0 \). Therefore, we must solve the following system of equations: 1. \( -x + z = 0 \) (the power of \( M \) must be 0), 2. \( 3x + y + 2z = 1 \) (the power of \( L \) must be 1), 3. \( -2x - y - z = 0 \) (the power of \( T \) must be 0). Step 3: Solving the System of Equations From equation (1), \( -x + z = 0 \), we get: \[ z = x \] Substitute this into equation (3): \[ -2x - y - x = 0 \quad \Rightarrow \quad -3x - y = 0 \quad \Rightarrow \quad y = -3x \] Now substitute \( z = x \) and \( y = -3x \) into equation (2): \[ 3x + (-3x) + 2x = 1 \quad \Rightarrow \quad 2x = 1 \quad \Rightarrow \quad x = \frac{1}{2} \] Thus, \( x = \frac{1}{2} \). Using \( x = \frac{1}{2} \), we find: \[ y = -3x = -\frac{3}{2}, \quad z = x = \frac{1}{2} \] Thus, the values of \( x \), \( y \), and \( z \) are: \[ x = \frac{1}{2}, \quad y = -\frac{3}{2}, \quad z = \frac{1}{2} \] Step 4: Conclusion Based on the calculations, the correct answer is: \[ \boxed{(C) \, y = \frac{1}{2}, z = \frac{3}{2}} \]